lineare transfinite Mengen. Berichte
(1911)

Tools

"... This paper is the first in a series whose objective is to study notions of large sets in the context of formal theories of constructivity. The two theories considered are Aczel's constructive set theory (CZF) and Martin-Lof's intuitionistic theory of types. This paper treats Mahlo&apos ..."

This paper is the first in a series whose objective is to study notions of large sets in the context of formal theories of constructivity. The two theories considered are Aczel&apos;s constructive set theory (CZF) and Martin-Lof&apos;s intuitionistic theory of types. This paper treats Mahlo&apos;s -numbers which give rise classically to the enumerations of inaccessibles of all transfinite orders. We extend the axioms of CZF and show that the resulting theory, when augmented by the tertium non datur, is equivalent to ZF plus the assertion that there are inaccessibles of all transfinite orders. Finally the theorems of that extension of CZF are interpreted in an extension of Martin-Lof&apos;s intuitionistic theory of types by a universe. 1 Prefatory and historical remarks The paper is organized as follows: After recalling Mahlo&apos;s -numbers and relating the history of universes in Martin-Lof type theory in section 1, we study notions of inaccessibility in the context of Aczel&apos;s constructive set theo...

"... This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meet ..."

This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meeting of the Association for Symbolic Logic at Haifa, in the Massachusetts Institute of Technology logic seminar, and to the Paris Logic Group. The author would like to express his thanks to the various organizers, as well as his gratitude to the Hebrew University of Jerusalem for its hospitality during the preparation of this article in the autumn of 1995.

...there can be no continuous bijection of any R n onto R m for msn, with Cantor [1879] himself providing an argument. As topology developed, the stress brought on by the lack of firm ground led Brouwer =-=[1911] to defini-=-tively establish the invariance of dimension in a seminal paper for algebraic topology. 12. This is emphasized by Hallett [1984] as Cantor's &quot;finitism.&quot; 48 AKIHIRO KANAMORI 13. After describ...

...epresentation systems. Such systems are by no means cooked up or impenetrable. As a rule, they utilize and extend wellknown set-theoretic hierarchies, for instance Mahlo's - and ae-number hierarchies =-=[27]-=-. 3.2 Ordinal functions based on a weakly inaccessible cardinal KPi is a set theory which originates from Kripke-Platek set theory and in addition has an axiom which says that any set is contained in ...

...hat the class of inaccessible cardinals is definably stationary, i.e., every definable closed unbounded class of ordinals contains an inaccessible cardinal. At the beginnings of set theory Paul Mahlo =-=[46, 47, 48]-=- had studied what are now known as the weakly Mahlo cardinals, those cardinals κ such that the set of smaller regular cardinals is stationary in κ, i.e., every closed unbounded subset of κ contains a ...

"... Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions and gauging their consistency strength. But set theory is also distinguished by having begun int ..."

Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions and gauging their consistency strength. But set theory is also distinguished by having begun intertwined with pronounced metaphysical attitudes, and these have even been regarded as crucial by some of its great developers. This has encouraged the exaggeration of crises in foundations and of metaphysical doctrines in general. However, set theory has proceeded in the opposite direction, from a web of intensions to a theory of extension par excellence, and like other fields of mathematics its vitality and progress have depended on a steadily growing core of mathematical proofs and methods, problems and results. There is also the stronger contention that from the beginning set theory actually developed through a progression of mathematical moves, whatever and sometimes in spite of what has been claimed on its behalf. What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been retained and developed by the current theory. The